Rationalization of Surds

 Rationalization of surds is very similar to rationalizing complex Numbers. Instead of having complex numbers in Fractions we have surds. Like conjugate for a Complex Number is evaluated, the same way we need to find the conjugate surd resolve the fraction. For instance, if we have a surd of form (x + √y) then corresponding conjugate for it will be (x - √y). Like we used to get real quantity in denominator of the fraction, here also we would be doing the same thing as it is difficult to divide by a surd. Let us learn the method for rationalizing surds.

Suppose we have a surd fraction as: (x + √y) / (h + √k). To resolve this division we 1st calculate the conjugate for denominator surd (h + √k) i.e. (h – √k). When we rationalize surds using the conjugate of denominator (h + √k), we get:
(x + √y) / (h + √k) = ((x + √y) / (h + √k)) * (((h - √k) / (h - √k)),
= (x h + √y k + (h √y– x √k)) / (h2 + k),

Where, (√k) 2 = k. Thus we get a real denominator which can now be distributed in numerator as follows:
(x + √y) / (h + √k) = (x h + √y k) / (h² + k) + (h √y– x √k)) / (h2 + k),
In case we have a surd fraction as: (x + √y) /k, where, 'k' belongs to real number line. In this surd fraction we do not have to do Rationalization as denominator of the fraction is real. Directly we distribute the denominator among the numerator to get a new surd:

(x + √y) /k = x /k + (√y /k).
This is all about rationalize surds.